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Higher-order Derivatives

The first derivative is the slope of a curve, and the second derivative is the slope of the slope, like acceleration. So what's the third derivative? (Fun fact: it's actually called jerk.)

Characteristics of f, f', f''

         

Let \(f\) be a function such that \(f(0)=50.\) If the above graph represents the graph of the first derivative of \(f\) on the interval \(0<x<3,\) which of the following is the correct description of \(f\) on this interval?

For the function \(y=x^3+x^2-x-1,\) determine the interval where \(y\) is decreasing.

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Imgur

Suppose \(f\) is a function such that \(f(0)=33\) and the first derivative of \(f\) on the interval \(-2 < x < 2\) is as shown in the above diagram. Which of the following is the correct description of the function \(f\) on this interval?

Imgur

Imgur

Suppose \(f\) is a function defined on the closed interval \(-3 \le x \le 4\) with \(f(0)=42,\) such that the graph of \(f',\) the derivative of \(f,\) on the interval is as shown in the above diagram. On what intervals is \(f\) increasing?

Imgur

Imgur

Suppose \(f\) is a function defined on the closed interval \(-3 \le x \le 4\) with \(f(0)=3\) such that the graph of \(f',\) the derivative of \(f,\) on the interval is as shown in the above diagram. Find the \(x\)-coordinates of the points of inflection of \(f.\)

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